We assume without loss of generality that the momentum vanishes, \(P = 0\) .
Let us observe that since \(\Vert u(t)\Vert _{H^{m+1}}\)  on a given time interval controls \([\rho ]_1\)  via the continuity equation, the local solution to (REF ) according to our criterion can be extended up to the critical time \(t = t^*\) , at which
\(\Vert u(t^*)\Vert _{H^{m+1}} = 2R,\) 
for the first time, i.e.
\(\Vert u(t)\Vert _{H^{m+1}} < 2R, \qquad t<t^*.\) 
Our goal will be to show that such time \(t^*\)  never happens, and thus the solution is global.
Lemma 4.3 
On the same time interval \([0,t^*]\)  we have
\(\frac{1}{2R} \le \underline{\rho }\le \overline{\rho }\le 2R.\) 

Let \(t^{**}\)  be the first time when one of these inequalities fails. We would like to show that \(t^{**} > t^*\) . If not, let us make a preliminary alignment estimate on the shorter time interval \([0,t^{**}]\) .
Recall the energy law (REF ). Note that
\(\phi (x,y) \ge \frac{\lambda }{M^{\tau /n}} {{1}}_{|x-y|<r_0}.\) 
So, we have
\(\frac{\mbox{d\,\,}}{\mbox{d}t}\mathcal {E}\le - \frac{c}{R^{2+\frac{\tau }{n}}} \int _{|x-y|<r_0} | u(x) - u(y)|^2 \, \mbox{d}y\, \mbox{d}x.\) 
Applying [1]}, we further continue
\(\frac{\mbox{d\,\,}}{\mbox{d}t}\mathcal {E}\le -\frac{c}{R^{2+\frac{\tau }{n}}} \int _{n} |u(x) - \bar{u}|^2 \, \mbox{d}x,\) 
where \(\bar{u}\)  is the usual average of \(u\) . Note that it may not be 0 despite vanishing momentum. Let us reinsert the density noting that \(\rho /R \le 2\) ,
\(\frac{\mbox{d\,\,}}{\mbox{d}t}\mathcal {E}\le -\frac{c}{R^{3+\frac{\tau }{n}}} \int _{n} \rho (x) |u(x) - \bar{u}|^2 \, \mbox{d}x.\) 
Expanding the square and using vanishing of the momentum we obtain
\(\frac{\mbox{d\,\,}}{\mbox{d}t}\mathcal {E}\le - \frac{c}{R^{3+\frac{\tau }{n}}} \int _{n} \rho (x) |u(x)|^2 \, \mbox{d}x= - \frac{c}{R^{3+\frac{\tau }{n}}} \mathcal {E}.\) 
Thus, on the time interval \([0,t^{**}]\)  we have
\(\mathcal {E}(t) \le \mathcal {E}_0 e^{- \frac{c}{R^a} t}, \qquad a = 3+\frac{\tau }{n}.\) 
Let us note that initial energy is bounded by (again using the vanishing momentum)
\(\mathcal {E}_0 \le \int _{{2n}} \rho _0(x)\rho _0(y) | u_0(x) - u_0(y)|^2 \, \mbox{d}y\, \mbox{d}x\le \mathcal {A}_0 M^2 \le C R^{2-N}.\) 
So,
\(\mathcal {E}(t) \le C R^{2-N}e^{- \frac{c}{R^a} t}.\) 
Now we estimate the decay of the amplitude \(\mathcal {A}\)  itself. Let us pick one coordinate of \(u\)  and evaluate at a point of its maximum \(x_+\) :
\(\begin{split}\frac{\mbox{d\,\,}}{\mbox{d}t}u(x_+) & \le \frac{c}{R^{\tau /n}} \int _{|z| < r_0} \rho (x_+ +z) [u(x_++z) - u(x_+)] \, \mbox{d}z\le \frac{c}{R^{\tau /n}} M^{1/2} \mathcal {E}^{1/2} - \frac{c}{R^{\tau /n}} u(x_+) \\&\le C R^{5/2 - \tau /n - N} e^{- \frac{c}{R^a} t} - \frac{c}{R^{\tau /n}} u(x_+).\end{split}\) 
Similarly,
\(\frac{\mbox{d\,\,}}{\mbox{d}t}u(x_-) \ge - C R^{5/2 - \tau /n - N} e^{- \frac{c}{R^a} t} + \frac{c}{R^{\tau /n}} u(x_-).\) 
Subtracting the two,
\(\frac{\mbox{d\,\,}}{\mbox{d}t}\mathcal {A}\le C R^{5/2 - \tau /n - N} e^{- \frac{c}{R^a} t} - \frac{c}{R^{\tau /n}} \mathcal {A}.\) 
By the Grönwall inquality,
\(\mathcal {A}(t) \le \mathcal {A}_0 e^{- \frac{c}{R^{\tau /n} }t} + C R^{5/2 - \tau /n - N} t e^{- \frac{c}{R^a} t} \le \frac{1}{R^N} e^{- \frac{c}{R^a} t} +C R^{5/2 - \tau /n - N+a} e^{- \frac{c}{R^a} t}.\) 
Thus,
\(\mathcal {A}(t) \le \frac{1}{R^{N-b}} e^{- \frac{c}{R^a} t} ,\) 
where \(b\)  depends only on \(n, \tau \) . By interpolation,
\(\Vert  \nabla u\Vert _\infty \le \mathcal {A}^{\theta } \Vert  u\Vert _{H^{m+1}}^{1-\theta } \le \frac{1}{R^{\theta (N-b) + \theta - 1}} e^{- \frac{\theta c}{R^a} t} .\) 
Integrating the continuity equation along characteristics, we obtain
\(\overline{\rho }\le \Vert  \rho _0 \Vert _\infty \exp \left\lbrace  \int _0^t \Vert  \nabla u\Vert _\infty \, \mbox{d}s\right\rbrace  \le R \exp \left\lbrace  \frac{1}{\theta c R^{\theta (N-b) + \theta - 1 - a }} \right\rbrace ,\) 
and similarly,
\(\underline{\rho }\ge \frac{1}{R} \exp \left\lbrace  - \frac{1}{\theta c R^{\theta (N-b) + \theta - 1 - a }} \right\rbrace .\) 
Clearly, if \(R\)  and \(N\)  are large enough the exponential is \(<2\) . This leads to a contradiction with the definition of \(t^{**}\) .
From this point on we will denote by \(\mathcal {N}(t)\)  any “negligent" quantity which has a bound of the form
\(\mathcal {N}(t) \le \frac{C}{R^{\theta N}} e^{ - \frac{c}{R^a} t },\) 
where \(C,c>0\) , and \(0<\theta <1\)  and \(a>0\)  depend only on \(\tau \) , \(n\) , \(\alpha \) , \(m\) , the parameters of the system.
We observe the identities
\(R^b \mathcal {N}\sim \mathcal {N}, \quad \mathcal {N}^\lambda \sim \mathcal {N}, \quad \mbox{etc}.\) 
As a consequence of the proof of the lemma we have shown that as long as (REF ) holds, the estimate on the amplitude (REF ) holds. As a result, by interpolation, we have similar exponential bounds in Hölder classes,
\([u]_1, [u]_2, [u]_3 \le \mathcal {N},\) 
and as a consequence, from the continuity equation,
\([\rho ]_1, [\rho ]_2 \le 2R,\) 
for all \(t<t^*\) , provided \(N\)  is large enough. As a further consequence, we obtain
\(\Vert e\Vert _\infty \le [u]_1+ \Vert  \mathcal {L}_{\phi } \rho \Vert _\infty .\) 
We appeal to [2]} (with \(r=1\)  and \(\gamma =0\) ) to conclude
\(\Vert  \mathcal {L}_\phi \rho \Vert _\infty \le R^b ([\rho ]_2 + \Vert \rho \Vert _\infty + [\rho ]_1^2) \le R^b,\) 
where \(b\)  depends only on the parameters of the system. Thus,
\(\Vert e(t)\Vert _\infty \le C R^b, \qquad t<t^*.\) 
Now, if we look back at the \(e\) -equation, we can see that all the transport terms contain a power of \([u]_1\) , and with (REF ) can be estimated by \(\le \mathcal {N}Y_m\) . At the same time the topological terms \(\mathcal {T}_1,\mathcal {T}_2\)  will include a power of \([u]_1\)  as a result of application of Lemma REF ,
\(\Vert \mathcal {T}_{1,2}\Vert  \le \mathcal {N}\Vert  \rho \Vert _{H^{m+\alpha }}^{\theta _1} \Vert  u\Vert _{H^{m+\alpha }}^{\theta _2} \le \mathcal {N}Y_m + \mathcal {N}+ \frac{\varepsilon }{R^{1+\tau /n}} \Vert u\Vert ^2_{H^{m+1+\frac{\alpha }{2}}}.\) 
Similarly, all paraproduct estimates of Lemma REF  for \(\mathcal {T}_3,\mathcal {T}_4\)  will include either \( \Vert  u\Vert _{H^{m+\alpha }}\)  or \(\Vert  \rho \Vert _{H^{m+\alpha }}\)  coupled with a power of \([u]_1\) . Again, in the latter case this results in a factor of \(\mathcal {N}\) , while in the former case, by interpolation
\(\Vert  u\Vert _{H^{m+\alpha }} \le \mathcal {N}\Vert u\Vert _{H^{m+1+\frac{\alpha }{2}}}^\theta .\) 
In summary we have a factor of \(\mathcal {N}\)  to appear in the main term of the \(e\) -equation:
\(\frac{\mbox{d\,\,}}{\mbox{d}t}\Vert e\Vert _{H^m}^2 \le \mathcal {N}Y_m + \mathcal {N}+ \frac{\varepsilon }{R^{1+\tau /n}} \Vert u\Vert ^2_{H^{m+1+\frac{\alpha }{2}}}.\) 
Examining the \(u\) -equation in a similar manner we conclude
\(\frac{\mbox{d\,\,}}{\mbox{d}t}\Vert u\Vert _{H^{m+1}}^2 \le \mathcal {N}Y_m + \mathcal {N}- \frac{c_0}{R^{1+\tau /n}} \Vert u\Vert ^2_{H^{m+1+\frac{\alpha }{2}}}.\) 
Adding the two together, we obtain
\(\frac{\mbox{d\,\,}}{\mbox{d}t}(Y_m+1) \le \mathcal {N}(Y_m +1) - \frac{c_0}{2R^{1+\tau /n}} \Vert u\Vert ^2_{H^{m+1+\frac{\alpha }{2}}}.\) 
Ignoring the dissipation term for a moment we conclude by integration that
\((Y_m+1)(t^*) \le (R+1) e^{R^{-\theta N}} \le 2R,\) 
if \(N\)  and \(R\)  are large enough. Thus,
\(Y_m(t^*) \le 3R.\) 
Plugging this back into (REF ) we conclude that at the critical time \(t^*\) ,
\(\frac{\mbox{d\,\,}}{\mbox{d}t}\Vert u\Vert _{H^{m+1}}^2 \le 3 \mathcal {N}R + \mathcal {N}- c_0 R^{1 - \tau /n} \le 4 R^{1-\theta N} - c_0 R^{1 - \tau /n} < 0,\) 
if again \(R\)  and \(N\)  are chosen large enough. This is a contradiction with the definition of \(t^*\) .
The flock convergence (REF ) follows immediately from the continuity equation and exponential decay of all norms of \(u\)  up to \(H^{m+1}\) .
